Just render shadows and reflections
You could use an Ambient Occlusion Shader for the plane. This will give the plane a constant color while still giving you the desired ambient shadows without the shadows from the light source. By default the Ambient Occlusion is pretty weak. We can use the compositor to make the Ambient Occlusion more prominent.


Add an Ambient Occlusion Shader to the plane
Enable the AO render pass


In the compositor:


use the RGB Curves or a similar node to make the AO pass darker
Invert: needed to properly control the factor input
Use the Multiply node to control the strength of the darkening
Subtract: this is a Color -> Mix node. This is the final output





NOTE: you can use the Distance parameter in the World Panel to change the proximity in which Ambient Occlusion is shown.



AO before vs. after compositing